This is actually a good way to tell someone is a narcissist. If someone spreads rumors at work or other situations it is almost certain they are narcissistic and petty. Note the more petty someone is the more narcissistic they are.
It is one thing to tell people about your past and generalize how people effected you without naming names but a narcissist will make sure to name the person they want to slander especially if they know you are friends with that person.
A couple of ways to test a covert narcissist is to tell them no to something they really want. If they react negatively it’s a bad sign. Also, make sure to ask them questions if they say something that sounds like it’s made up. If they get offended you are questioning their story that is another red flag.
I went through this type of slander in my last year of school when a girl I suspect was a malignant narcissist asked me out. When I told her no she threatened to spread a rumor to make sure no one else would go out with me. Sure enough she spread some rumor that was obviously a lie since nothing bad ever happened between us yet people believed it to such an extent that a bus driver almost refused to drive me home.
I had to get my parents involved to get the driver to back down. To this day I still don’t know what rumor she spread but I wish there were laws that allowed people to file lawsuits even for past acts of slander since she basically got away with what she did. If someone slanders you make sure to collect evidence and statements from as many people as possible.